Where've I heard something similar to that before...? Oh, yeah.
Animals can most certainly get stressed; there are any number of articles & studies that prove this.
stress in animals
Unfortunate as it may be, it's incumbent upon you to educate them. It's
your chameleon, and you're responsible for it in every way. If they don't agree or like it, well... I can't tell you what to do, but I think you know or can figure it out. You know your family better than anyone here.
My observations or suggestions:
1. All chameleons (or reptiles or animals) are different and individuals, so what applies to one male panther may not necessarily apply to another. We have to learn to read or charges, which takes some studying (i.e. watching) but not to the point of freaking them out.
2. Learn all the indicators & signs of stress for your species, and eventually, for your individual. This is usually communicated by color or color changes, but there may be other body language or behavioral indicators as well. This can include (but not limited to) changes in eating, drinking, basking, pooping, etc. A journal can help.
3. A piece of advice I've oft heard... "Pretend you don't see them"—at least until they've learned you don't intend them harm, won't eat them, etc., which (I agree) has to be on
their terms—not yours. They'll be watching (for) your eyes (and anyone else's).
4. Personally, I'd introduce strangers or family members
one at a time—
not en masse. Each time, try to get a read on the cham's feelings/reactions. They can feel differently about different individuals just as we (or other animals—a cat or dog) can. Take it slow.